RDP 9812: An Optimising Model for Monetary Policy Analysis: Can Habit Formation Help? Appendix A: First-order Conditions for the Nonlinear Model

Beginning with the definition of period utility:

the overall utility function:

and the habit-formation reference consumption level:

The derivative of U with respect to Ct is:

Noting that Inline Equation that Inline Equation and that Inline Equation then we can express:

which collapses to a more compact discounted summation:


then we have the derivatives of utility U with respect to Ct and Ct+1

Combining these with a standard budget constraint (with time-varying real interest rate rt) in a Lagrangian, we obtain the first-order conditions:

Both of these expressions must equal zero for an optimum, yielding an Euler equation: